@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr: a skos:ConceptScheme .
psr:-KP6XX270-0
  skos:definition """La fonction tangente hyperbolique, notée <b>tanh</b> (ou th) est la fonction complexe suivante&nbsp;:
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\egin{matrix}\\	anh :&amp;\\\\mathbb {C} \\\\setminus {\\
m {i}}\\\\pi \\\\left(\\\\mathbb {Z} +{\\\\dfrac {1}{2}}\\
ight)&amp;\\\\longrightarrow &amp;\\\\mathbb {C} \\\\\\\\&amp;z&amp;\\\\longmapsto &amp;{\\\\dfrac {\\\\sinh(z)}{\\\\cosh(z)}}\\\\end{matrix}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mtable rowspacing="4pt" columnspacing="1em">
<br/>            <mtr>
<br/>              <mtd>
<br/>                <mi>tanh</mi>
<br/>                <mo>:</mo>
<br/>              </mtd>
<br/>              <mtd>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi mathvariant="double-struck">C</mi>
<br/>                </mrow>
<br/>                <mo class="MJX-variant">∖<!-- ∖ --></mo>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">i</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mi>π<!-- π --></mi>
<br/>                <mrow>
<br/>                  <mo>(</mo>
<br/>                  <mrow>
<br/>                    <mrow class="MJX-TeXAtom-ORD">
<br/>                      <mi mathvariant="double-struck">Z</mi>
<br/>                    </mrow>
<br/>                    <mo>+</mo>
<br/>                    <mrow class="MJX-TeXAtom-ORD">
<br/>                      <mstyle displaystyle="true" scriptlevel="0">
<br/>                        <mfrac>
<br/>                          <mn>1</mn>
<br/>                          <mn>2</mn>
<br/>                        </mfrac>
<br/>                      </mstyle>
<br/>                    </mrow>
<br/>                  </mrow>
<br/>                  <mo>)</mo>
<br/>                </mrow>
<br/>              </mtd>
<br/>              <mtd>
<br/>                <mo stretchy="false">⟶<!-- ⟶ --></mo>
<br/>              </mtd>
<br/>              <mtd>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi mathvariant="double-struck">C</mi>
<br/>                </mrow>
<br/>              </mtd>
<br/>            </mtr>
<br/>            <mtr>
<br/>              <mtd></mtd>
<br/>              <mtd>
<br/>                <mi>z</mi>
<br/>              </mtd>
<br/>              <mtd>
<br/>                <mo stretchy="false">⟼<!-- ⟼ --></mo>
<br/>              </mtd>
<br/>              <mtd>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mstyle displaystyle="true" scriptlevel="0">
<br/>                    <mfrac>
<br/>                      <mrow>
<br/>                        <mi>sinh</mi>
<br/>                        <mo>⁡<!-- ⁡ --></mo>
<br/>                        <mo stretchy="false">(</mo>
<br/>                        <mi>z</mi>
<br/>                        <mo stretchy="false">)</mo>
<br/>                      </mrow>
<br/>                      <mrow>
<br/>                        <mi>cosh</mi>
<br/>                        <mo>⁡<!-- ⁡ --></mo>
<br/>                        <mo stretchy="false">(</mo>
<br/>                        <mi>z</mi>
<br/>                        <mo stretchy="false">)</mo>
<br/>                      </mrow>
<br/>                    </mfrac>
<br/>                  </mstyle>
<br/>                </mrow>
<br/>              </mtd>
<br/>            </mtr>
<br/>          </mtable>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\egin{matrix}\\	anh :&amp;\\\\mathbb {C} \\\\setminus {\\
m {i}}\\\\pi \\\\left(\\\\mathbb {Z} +{\\\\dfrac {1}{2}}\\
ight)&amp;\\\\longrightarrow &amp;\\\\mathbb {C} \\\\\\\\&amp;z&amp;\\\\longmapsto &amp;{\\\\dfrac {\\\\sinh(z)}{\\\\cosh(z)}}\\\\end{matrix}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a826e28abc9dd820afacf45f7130c8d7323214" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -5.838ex; width:41.655ex; height:12.843ex;" alt="{\\\\displaystyle {\\egin{matrix}\\	anh :&amp;\\\\mathbb {C} \\\\setminus {\\
m {i}}\\\\pi \\\\left(\\\\mathbb {Z} +{\\\\dfrac {1}{2}}\\
ight)&amp;\\\\longrightarrow &amp;\\\\mathbb {C} \\\\\\\\&amp;z&amp;\\\\longmapsto &amp;{\\\\dfrac {\\\\sinh(z)}{\\\\cosh(z)}}\\\\end{matrix}}}"></span></dd></dl>
<br/>où <span class="texhtml">sinh</span> est la fonction sinus hyperbolique et <span class="texhtml">cosh</span> la fonction cosinus hyperbolique. 
<br/>Cette définition est analogue à celle de la fonction tangente comme rapport du sinus et du cosinus, et d'ailleurs, on a (pour tous les <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle z}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>z</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle z}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="z"></span> du domaine de définition) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\	anh(z)=-i\\	an(iz)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>tanh</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>i</mi>
<br/>        <mi>tan</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>i</mi>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\	anh(z)=-i\\	an(iz)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/062a5f7e769739355dac92bd1243db50b9a42677" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.705ex; height:2.843ex;" alt="{\\\\displaystyle \\	anh(z)=-i\\	an(iz)}"></span>, ou encore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\	an(z)=-i\\	anh(iz)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>tan</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>i</mi>
<br/>        <mi>tanh</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>i</mi>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\	an(z)=-i\\	anh(iz)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70612a8a34f5032cf7eceefd62c18fac46e82317" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.705ex; height:2.843ex;" alt="{\\\\displaystyle \\	an(z)=-i\\	anh(iz)}"></span> pour tout <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle z\\\\in \\\\mathbb {C} \\\\setminus \\\\pi \\\\left(\\\\mathbb {Z} +{\\rac {1}{2}}\\
ight)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>z</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">C</mi>
<br/>        </mrow>
<br/>        <mo class="MJX-variant">∖<!-- ∖ --></mo>
<br/>        <mi>π<!-- π --></mi>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="double-struck">Z</mi>
<br/>            </mrow>
<br/>            <mo>+</mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mn>1</mn>
<br/>                <mn>2</mn>
<br/>              </mfrac>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle z\\\\in \\\\mathbb {C} \\\\setminus \\\\pi \\\\left(\\\\mathbb {Z} +{\\rac {1}{2}}\\
ight)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49339561a6f5ec4d820c64fe0e76f1ef9d81758f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:19.331ex; height:6.176ex;" alt="{\\\\displaystyle z\\\\in \\\\mathbb {C} \\\\setminus \\\\pi \\\\left(\\\\mathbb {Z} +{\\rac {1}{2}}\\
ight)}"></span>.
<br/>La tangente hyperbolique peut s'exprimer à l'aide de la fonction exponentielle&nbsp;:
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\	anh(z)={\\rac {{\\
m {e}}^{z}-{\\
m {e}}^{-z}}{{\\
m {e}}^{z}+{\\
m {e}}^{-z}}}={\\rac {{\\
m {e}}^{2z}-1}{{\\
m {e}}^{2z}+1}}={\\rac {1-{\\
m {e}}^{-2z}}{1+{\\
m {e}}^{-2z}}}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>tanh</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>z</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>−<!-- − --></mo>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mi>z</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>z</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>+</mo>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mi>z</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                  <mi>z</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                  <mi>z</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mn>1</mn>
<br/>              <mo>−<!-- − --></mo>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mn>2</mn>
<br/>                  <mi>z</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mn>1</mn>
<br/>              <mo>+</mo>
<br/>              <msup>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="normal">e</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mn>2</mn>
<br/>                  <mi>z</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\	anh(z)={\\rac {{\\
m {e}}^{z}-{\\
m {e}}^{-z}}{{\\
m {e}}^{z}+{\\
m {e}}^{-z}}}={\\rac {{\\
m {e}}^{2z}-1}{{\\
m {e}}^{2z}+1}}={\\rac {1-{\\
m {e}}^{-2z}}{1+{\\
m {e}}^{-2z}}}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af9916c40421267102260e89cc37a619dbcbe1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:43.184ex; height:6.176ex;" alt="\\	anh(z)={\\rac  {{{\\
m {e}}}^{z}-{{\\
m {e}}}^{{-z}}}{{{\\
m {e}}}^{z}+{{\\
m {e}}}^{{-z}}}}={\\rac  {{{\\
m {e}}}^{{2z}}-1}{{{\\
m {e}}}^{{2z}}+1}}={\\rac  {1-{{\\
m {e}}}^{{-2z}}}{1+{{\\
m {e}}}^{{-2z}}}}."> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Tangente_hyperbolique">https://fr.wikipedia.org/wiki/Tangente_hyperbolique</a>)"""@fr ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:broader psr:-R92BT00M-4 ;
  skos:prefLabel "hyperbolic tangent"@en, "tangente hyperbolique"@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Tangente_hyperbolique> .

psr:-R92BT00M-4
  skos:prefLabel "hyperbolic function"@en, "fonction hyperbolique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-KP6XX270-0 .